Can AI write good jokes yet?
I don't think the AI has gotten to that point yet. Here are some of the interesting papers on the subject:


A paper was recently written that attempted to generate jokes using unsupervised learning. The jokes are formulaic: they're all of the form "I like my X like I like my Y: Z" where X and Y are nouns, and Z is an adjective that can describe both X and Y. Here are some of the jokes generated in this paper: 

I like my relationships like I like my source, open
I like my coffee like I like my war, cold
I like my boys like I like my sectors, bad


How funny these jokes are is a matter of personal taste I guess. 
Another paper by Dario Bertero and Pascale Fung makes use of an LSTM to predict humor from a dataset of the Big Bang theory shows. This is not generating jokes but finding out where the jokes are said in this dataset (so theoretically, the resulting labelled dataset can hopefully be used to train a model to create jokes). 
Yet another paper is that by He Ren, Quan Yang. Unlike the first paper mentioned above which was unsupervised, this is a supervised learning model. Their neural network model, generates jokes such as: 

Apple is teaming up with Playboy Magazine in the self driving office.
One of the top economy in China , Lady Gaga says today that Obama is legal.
Google Plus has introduced the remains that lowers the age of coffee.
According to a new study , the governor of film welcome the leading actor of Los Angeles area , Donald Trump .



My two cents: 

As of this writing, it appears that Multi-layer Recurrent Neural Networks (LSTM, GRU, RNN) for character-level language models are by far the most promising way to go about it. Maybe if you find some really cool data you can come up with some funny jokes, similar to how Janelle Shane was able to generate what I find to be really funny pickup lines such as:

Are you a 4loce? Because you’re so hot!
I want to get my heart with you.
You are so beautiful that you know what I mean.
I have a cenver? Because I just stowe must your worms.
Hey baby, I’m swirked to gave ever to say it for drive. 
If I were to ask you out?
You must be a tringle? Cause you’re the only thing here.
I’m not on your wears, but I want to see your start.
You are so beautiful that you make me feel better to see you.
Hey baby, you’re to be a key? Because I can bear your toot?
I don’t know you.
I have to give you a book, because you’re the only thing in your eyes.
Are you a candle? Because you’re so hot of the looks with you.
I want to see you to my heart.
If I had a rose for every time I thought of you, I have a price tighting.
I have a really falling for you.
Your beauty have a fine to me.
Are you a camera? Because I want to see the most beautiful than you.
I had a come to got your heart.
You’re so beautiful that you say a bat on me and baby.
You look like a thing and I love you.
Hello.